2014/02/03 by Donald M. Davis, Davis, Donald M.
Mathematics · #05A99 #11B73 #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #advanced mathematical theories #math.CO #math.NT #msc:05A99 #msc:11B73
paper · pdf · doi:10.48550/arxiv.1402.0433
25 pages
arxiv created 2014/02/03 · openalex publication_date 2014/02/03 · arxiv updated 2014/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Pn(x)=\frac1n!∑\binom n2i+1(2i+1)x. This extends to a continuous function on the 2-adic integers, the nth 2-adic partial Stirling function. We show that (-1)n+1Pn is the only 2-adically continuous approximation to S(x,n), the Stirling number of the second kind. We present extensive information about the zeros of Pn, for which there are many interesting patterns. We prove that if e≥2 and 2e+1≤ n≤ 2e+4, then Pn has exactly 2e-1 zeros, one in each mod 2e-1 congruence. We study the relationship between the zeros of P2e+Δ and PΔ, for 1≤Δ≤ 2e, and the convergence of P2e+Δ(x) as e→∞.